Enigma was originally broken by a team of Polish cryptanalysts, who invented both a paper method to break it (Zygalski sheets) and the machine shown in movie (Rejewski's bomba). Turing later worked on a team to improve the bomba, which was a significant achievement but he didn't invent or spearhead the whole thing.
All this "don't use a machine to do a man's job" was bullshit invented for the movie, as were like half the other conflicts. Everyone involved knew that breaking Enigma was important, and they were already using a machine to do it, but after the Nazis changed their use of Enigma, the Brits needed mathematical insights to improve the speed of their machine.
The main characters in the film did exist and had some of the personality traits shown, but mostly didn't interact they way that the film showed. Turing was openly gay. Cairncross (the spy) didn't work with Turing. The codebreakers didn't choose which intercepted information to use and how, etc.